The price of anarchy in games of incomplete information

We define smooth games of incomplete information. We prove an "extension theorem" for such games: price of anarchy bounds for pure Nash equilibria for all induced full-information games extend automatically, without quantitative degradation, to all mixed-strategy Bayes-Nash equilibria with respect to a product prior distribution over players' preferences. We also note that, for Bayes-Nash equilibria in games with correlated player preferences, there is no general extension theorem for smooth games.